How to copy 1 part of a mesh to other places on the same mesh (without mirroring)
When you separate the selection, you're actually creating a new object. It seems what you want to do is to split. You don't need to split the part since you have already duplicated it. If you want to &quot;cut&quot; it without copying, and keep it in the same object, what you need to do is right-click and choose Split &gt; Selection - Y is the standard keymap.
Once you duplicate the part, place it somewhere, then the easiest thing to move it where it is intended is to select vertex mode, then deselect and reselect a vertex on the duplicated part (while making sure that the part is still wholly selected), activate vertex snap, and move it to snap with the correct vertex at its intended location.